Optimal reinforcing networks for elastic membranes

Giovanni Alberti; Giuseppe Buttazzo; Serena Guarino Lo Bianco; Édouard Oudet

doi:10.3934/nhm.2019023

Article Contents

2019, Volume 14, Issue 3: 589-615. Doi: 10.3934/nhm.2019023

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Optimal reinforcing networks for elastic membranes

1.
Dipartimento di Matematica, Università di Pisa, l.go B. Pontecorvo 5, 56127 Pisa, Italy
2.
Dipartimento di Matematica e an, 80126 Napoli, Italy
3.
Laboratoire Jean Kuntzmann, Université Grenoble Alpes, 38041 Grenoble, France

^* Corresponding author
^* Corresponding author

Received: August 2018

Revised: March 2019

Published: September 2019

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Abstract

In this paper we study the optimal reinforcement of an elastic membrane, fixed at its boundary, by means of a network (connected one-dimensional structure), that has to be found in a suitable admissible class. We show the existence of an optimal network, and observe that such network carries a multiplicity that in principle can be strictly larger than one. Some numerical simulations are shown to confirm this issue and to illustrate the complexity of the optimal network when the total length becomes large.

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Mathematics Subject Classification: 49J45, 49Q10, 35R35, 35J25, 49M05.

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Figure 1. Approximation of globally optimal reinforcement structures for $ m = 0.5 $, $ L = 1, \, 2 $ and $ 3 $. The upper colorbar is related to the weights $ \theta $ which colors the optimal reinforcement set on the left, whereas the lower colorbar stands for the tangential gradient plotted on the connected set on the right picture

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Figure 3. Approximation of globaly optimal reinforcement structures for $ m = 0.5 $, $ L = 4, \, 5 $ and $ 6 $. The upper colorbar is related to the weights $ \theta $ which colors the optimal reinforcement set on the left, whereas the lower colorbar stands for the tangential gradient plotted on the connected set on the right picture

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Figure 2. Approximation of globaly optimal reinforcement strucutres for m = 0.5, L = 1.5, 2.5 and 5 for a source consisting of two dirac masses. The upper colorbar is related to the weights θ which colors the optimal reinforcement set on the left, whereas the lower colorbar stands for the tangential gradient plotted on the connected set on the right picture

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Table 1. Reinforcement values computed on a fine mesh of $ 10^6 $ elements for classical and computed connected sets for $ m = 0.5 $

Length constraint	Theoretical guesses	Computed optimal networks
1	-0.179471 (radius)	-0.178873
2	-0.165095 (diameter)	-0.161944
3	-0.152676 (star)	-0.149601
4	-0.141969 (cross)	-0.138076
5	-	-0.127661
6	-	-0.117140

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